Operator growth from global out-of-time-order correlators

In chaotic many-body systems, scrambling or the operator growth can be diagnosed by out-of-time-order correlators of local operators. We show that operator growth also has a sharp imprint in out-of-time-order correlators of global operators. In particular, the characteristic spacetime shape of growing local operators can be accessed using global measurements without any local control or readout. Building on an earlier conjectured phase diagram for operator growth in chaotic systems with power-law interactions, we show that existing nuclear spin data for out-of-time-order correlators of global operators are well fit by our theory. We also predict super-polynomial operator growth in dipolar systems in 3d and discuss the potential observation of this physics in future experiments with nuclear spins and ultra-cold polar molecules.

Formally, MQC can be defined as 64 g n = 1 tr(I 2 z ) tr(ρ n ρ −n ) lution to measure the Fourier transform of the multiple 66 quantum coherence.
I(φ, t) = 1 tr(I 2 z ) tr(e iφIz ρ(t)e iφIz e iHt I z e −iHt ) (9) In fact, expanding ρ(t) and using the property in Eq. (7), 68 we have Therefore if we sample I(φ, t) at discrete values of φ, where I is omitted later in the calculation, resulting in 118 the schematic ρ(t) ∼ Z(−t).

119
Since Z is a many-body operator, this expansion is 120 formally only valid if the system size is sufficiently small.

121
The gyro-magnetic ratio in     The density operator ρ can be decomposed as where ρ n increases the total spin z quantum number by n.

182
Operators like X and Y change the total spin z quantum and the "intensity" (25) operators that belong to the n-quantum coherence space.

198
If all these operators are equally likely, then g n will be 199 roughly a Gaussian function, g n ∼ exp(− n 2 K ). Therefore, 200 the second moment of g n , the global OTOC, will scale as 201 K, the number of spins in the system.

202
In the dynamical setting, this idea is generalized by Using the Pauli string basis, the number K here is the 244 number of Pauli operators in the string. In Sec. II A, 245 1 Other variants that take the distribution to be a superposition of Gaussian functions with different cluster sizes also predict exponential growth [13,14].
we introduced this number as the effective size of the the transition probability can be written as where N is the total number of spins. 260 We simulate this process and reproduce the multiple  OTOCs: We thus conclude that the global OTOC measures the 276 area under the local OTOC curve.

277
In this section, we present a more rigorous calculation 278 to show why the off-diagonal terms can be neglected for 279 the sake of estimating the global OTOC scaling.

280
In the following, we analyze the diagonal and off- We represent each term in a tensor network diagram as

328
There are four copies of the local Hilbert space in the 329 structure u⊗u * ⊗u⊗u * . We define two tensors in among 330 Hilbert spaces as where the graphical notation means a delta function be- such that + * |+ = − * |− = 1, and + * |− = − * |+ = 336 0. For states defined on two sites, the dual basis are We can similarly construct the kets ++| and −−| from 338 | and |, and so are their dual basis.

339
With these facilities, the random averaging of u ⊗ u * ⊗ 340 u ⊗ u * can be written as and at b/d as (39) We have write with the more general traceless operators 361 O a,b,c,d here.

362
The rules for the spin assignment is restrictive due to also not optimal, but we neglect this factor. Overall, the 457 diagram in Supplementary Figure 6 can be a factor of 458 q −(2v B t−x bd q −x bd = q −2v B t smaller than a local OTOC.

459
Even if there can be (2t) 2 terms, the contribution is neg-460 ligible than the v B t local OTOCs. We have boundary conditions We generalize the domain wall cost analysis to the off-483 diagonal terms when the interaction is long-ranged. 484 In the discussion of the local interaction, we see 485 that when b = d, the ⊥ cluster brings in a factor of 486 q −(v B t−x bd ) .

487
For case 2, this suppression factor can be generalized